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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

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At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Nicely Similar

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
triangle diagram
Well done to the people who found the height (48) and then used it to get the triangle side lengths, but special congratulations to Michael from Taradale High School for a more direct approach using the powerful property that this arrangement contains three similar triangles.

If the yellow triangle has a hypotenuse, h, and if the yellow-orange triangle and the yellow triangle are similar, then the ratio, h : 100 is equal to the ratio 36 : h

From that we know that h must be 60, and using the same thinking again the hypotenuse of the orange triangle must be 80.